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Equivalence of hybrid dynamical models
 AUTOMATICA
, 2001
"... This paper establishes equivalences among five classes of hybrid systems: mixed logical dynamical (MLD) systems, linear complementarity (LC) systems, extended linear complementarity (ELC) systems, piecewise affine (PWA) systems, and maxminplusscaling (MMPS) systems. Some of the equivalences are es ..."
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Cited by 115 (29 self)
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This paper establishes equivalences among five classes of hybrid systems: mixed logical dynamical (MLD) systems, linear complementarity (LC) systems, extended linear complementarity (ELC) systems, piecewise affine (PWA) systems, and maxminplusscaling (MMPS) systems. Some of the equivalences are established under (rather mild) additional assumptions. These results are of paramount importance for transferring theoretical properties and tools from one class to another, with the consequence that for the study of a particular hybrid system that belongs to any of these classes, one can choose the most convenient hybrid modeling framework.
Heemels: A Bayesian Approach to Identification of Hybrid Systems
 IEEE Trans. on Automatic Control
, 2005
"... Abstract—In this paper, we present a novel procedure for the identification of hybrid systems in the class of piecewise ARX systems. The presented method facilitates the use of available a priori knowledge on the system to be identified, but can also be used as a blackbox method. We treat the unkno ..."
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Cited by 42 (4 self)
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Abstract—In this paper, we present a novel procedure for the identification of hybrid systems in the class of piecewise ARX systems. The presented method facilitates the use of available a priori knowledge on the system to be identified, but can also be used as a blackbox method. We treat the unknown parameters as random variables, described by their probability density functions. The identification problem is posed as the problem of computing the a posteriori probability density function of the model parameters, and subsequently relaxed until a practically implementable method is obtained. A particle filtering method is used for a numerical implementation of the proposed procedure. A modified version of the multicategory robust linear programming classification procedure, which uses the information derived in the previous steps of the identification algorithm, is used for estimating the partition of the piecewise ARX map. The proposed procedure is applied for the identification of a component placement process in pickandplace machines. Index Terms—Hybrid systems, identification. I.
Lecture notes on hybrid systems
, 2004
"... The aim of this course is to introduce some fundamental concepts from the area of hybrid systems, that is dynamical systems that involve the interaction of continuous (real valued) states and discrete (finite valued) states. Applications where these types of dynamics play a prominent role will be hi ..."
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Cited by 20 (0 self)
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The aim of this course is to introduce some fundamental concepts from the area of hybrid systems, that is dynamical systems that involve the interaction of continuous (real valued) states and discrete (finite valued) states. Applications where these types of dynamics play a prominent role will be highlighted. We will introduce general methods for investigating properties such as existence of solutions, reachability and decidability of hybrid systems. The methods will be demonstrated on the motivating applications. Students who successfully complete the course should be able to appreciate the diversity of phenomena that arise in hybrid systems and how discrete “discrete ” entities and concepts such as automata, decidability and bisimulation can coexist with continuous entities and
On hybrid systems and closedloop MPC systems
 IEEE TRANSACTIONS ON AUTOMATIC CONTROL
, 2002
"... The following five classes of hybrid systems were recently proved to be equivalent: linear complementarity (LC) systems, extended linear complementarity (ELC) systems, mixed logical dynamical (MLD) systems, piecewise affine (PWA) systems, and maxminplusscaling (MMPS) systems. Some of the equivalen ..."
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Cited by 20 (4 self)
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The following five classes of hybrid systems were recently proved to be equivalent: linear complementarity (LC) systems, extended linear complementarity (ELC) systems, mixed logical dynamical (MLD) systems, piecewise affine (PWA) systems, and maxminplusscaling (MMPS) systems. Some of the equivalences were obtained under additional assumptions, such as boundedness of system variables. In this paper, for linear or hybrid plants in closedloop with a model predictive control (MPC) controller based on a linear model and fulfilling linear constraints on input and state variables, we provide a simple and direct proof that the closedloop system (clMPC) is a subclass of any of the former five classes of hybrid systems. This result opens the use of tools developed for hybrid systems (such as stability, robust stability, and safety analysis tools) to study closedloop properties of MPC.
Linear complementarity systems: Zeno states
 SIAM J. CONTROL OPTIM
, 2005
"... A linear complementarity system (LCS) is a hybrid dynamical system defined by a linear timeinvariant ordinary differential equation coupled with a finitedimensional linear complementarity problem (LCP). The present paper is the first of several papers whose goal is to study some fundamental issue ..."
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Cited by 20 (6 self)
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A linear complementarity system (LCS) is a hybrid dynamical system defined by a linear timeinvariant ordinary differential equation coupled with a finitedimensional linear complementarity problem (LCP). The present paper is the first of several papers whose goal is to study some fundamental issues associated with an LCS. Specifically, this paper addresses the issue of Zeno states and the related issue of finite number of mode switches in such a system. The cornerstone of our study is an expansion of a solution trajectory to the LCS near a given state in terms of an observability degree of the state. On the basis of this expansion and an inductive argument, we establish that an LCS satisfying the Pproperty has no strongly Zeno states. We next extend the analysis for such an LCS to a broader class of problems and provide sufficient conditions for a given state to be weakly nonZeno. While related modeswitch results have been proved by Brunovsky and Sussmann for more general hybrid systems, our analysis exploits the special structure of the LCS and yields new results for the latter that are of independent interest and complement those by these two and other authors.
Projected Dynamical Systems in a Complementarity Formalism
 Operations Research Letters
, 1999
"... Projected dynamical systems have been introduced by Dupuis and Nagurney as dynamic extensions of variational inequalities. In the systems and control literature, complementarity systems have been studied as input/output dynamical systems whose inputs and outputs are connected through complementarity ..."
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Cited by 18 (4 self)
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Projected dynamical systems have been introduced by Dupuis and Nagurney as dynamic extensions of variational inequalities. In the systems and control literature, complementarity systems have been studied as input/output dynamical systems whose inputs and outputs are connected through complementarity conditions. We show here that, under mild conditions, projected dynamical systems can be written as complementarity systems. Keywords: variational inequalities, complementarity, discontinuous dynamical systems, systems theory, optimization. 1 Introduction In this paper, we connect two classes of discontinuous dynamical systems. One is the class of projected dynamical systems introduced by Dupuis and Nagurney [4] and further developed by Nagurney and Zhang [14]. These systems are described by differential equations of the form x(t) = \Pi K (x(t); \GammaF (x(t))); (1) Dept. of Electrical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands, an...
Conewise linear systems: nonZenoness and observability
 SIAM J. Control Optim
"... Abstract. Conewise linear systems are dynamical systems in which the state space is partitioned into a finite number of nonoverlapping polyhedral cones on each of which the dynamics of the system is described by a linear differential equation. This class of dynamical systems represents a large numbe ..."
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Cited by 18 (7 self)
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Abstract. Conewise linear systems are dynamical systems in which the state space is partitioned into a finite number of nonoverlapping polyhedral cones on each of which the dynamics of the system is described by a linear differential equation. This class of dynamical systems represents a large number of piecewise linear systems, most notably, linear complementarity systems with the Pproperty and their generalizations to affine variational systems, which have many applications in engineering systems and dynamic optimization. The challenges of dealing with this type of hybrid system are due to two major characteristics: mode switchings are triggered by state evolution, and states are constrained in each mode. In this paper, we first establish the absence of Zeno states in such a system. Based on this fundamental result, we then investigate and relate several state observability notions: shorttime and Ttime (or finitetime) local/global observability. For the shorttime observability notions, constructive, finitely verifiable algebraic (both sufficient and necessary) conditions are derived. Due to their longtime modetransitional behavior, which is very difficult to predict, only partial results are obtained for the Ttime observable states. Nevertheless, we completely resolve the Ttime local observability for the bimodal conewise linear system, for finite T, and provide numerical examples to illustrate the difficulty associated with the longtime observability.
On the equivalence of classes of hybrid dynamical models
 40TH IEEE CONFERENCE ON DECISION AND CONTROL
, 2001
"... We establish equivalences among five classes of hybrid systems, that we have encountered in previous research: mixed logical dynamical systems, linear complementarity systems, extended linear complementarity systems, piecewise affine systems, and maxminplusscaling systems. These results are of pa ..."
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Cited by 16 (1 self)
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We establish equivalences among five classes of hybrid systems, that we have encountered in previous research: mixed logical dynamical systems, linear complementarity systems, extended linear complementarity systems, piecewise affine systems, and maxminplusscaling systems. These results are of paramount importance for transferring properties and tools from one class to another.
Lyapunov stability of complementarity and extended systems
 SIAM J. Optim
"... Abstract. A linear complementarity system (LCS) is a piecewise linear dynamical system consisting of a linear timeinvariant ordinary differential equation (ODE) parameterized by an algebraic variable that is required to be a solution to a finitedimensional linear complementarity problem (LCP), wh ..."
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Cited by 15 (4 self)
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Abstract. A linear complementarity system (LCS) is a piecewise linear dynamical system consisting of a linear timeinvariant ordinary differential equation (ODE) parameterized by an algebraic variable that is required to be a solution to a finitedimensional linear complementarity problem (LCP), whose constant vector is a linear function of the differential variable. Continuing the authors’ recent investigation of the LCS from the combined point of view of system theory and mathematical programming, this paper addresses the important systemtheoretic properties of exponential and asymptotic stability for an LCS with a C1 state trajectory. The novelty of our approach lies in our employment of a quadratic Lyapunov function that involves the auxiliary algebraic variable of the LCS; when expressed in the state variable alone, the Lyapunov function is piecewise quadratic, and thus nonsmooth. The nonsmoothness feature invalidates standard stability analysis that is based on smooth Lyapunov functions. In addition to providing sufficient conditions for exponential stability, we establish a generalization of the wellknown LaSalle invariance theorem for the asymptotic stability of a smooth dynamical system to the LCS, which is intrinsically a nonsmooth system. Sufficient matrixtheoretic copositivity conditions are introduced to facilitate the verification of the stability properties. Properly specialized, the latter conditions are satisfied by a passivelike LCS and certain hybrid linear systems having common quadratic Lyapunov functions. We provide numerical examples to illustrate the stability results. We also develop an extended local exponential stability theory for nonlinear complementarity systems and differential variational inequalities, based on a new converse theorem for ODEs with Bdifferentiable righthand sides. The latter theorem asserts that the existence of a “Bdifferentiable Lyapunov function ” is a necessary and sufficient condition for the exponential stability of an equilibrium of such a differential system.